2 PARTICULAR CASES AND EXAMPLES 251
and that there exists a positive nonzero G-invariant measure /u- on X. Let P (resp. C)
be a G -packing (resp. a G-covering) such that P and C are /n-integrable. Show that
ft(Q £ /A(P). (Remark that ^t(C) ^ £ p(C n j • P) = £ /^- r - C n P).)
seG se G
(b) Suppose that there exists a G-invariant distance d on X defining the topology
of X. Let A(G) denote the greatest lower bound of the numbers /u,(C), where C runs
through all integrable G-coverings of X. Let r be a real number >0 such that there
exists a point a e X for which ju,(B(#; r)) > A(G). Show that there exists s ^ e in G
such that d(a, s - a) < 2r.
(c) Suppose that X is a locally compact group, /x a left Haar measure on X, and G a
denumerable subgroup acting on X by left translations. Show that if A is an integrable
subset of X such that fju(A) > A(G), then there exists s e G n AA"1 such that s ^ e.
(d) With the same hypotheses as in (a), let F be a jie-integrable G -tessellation and let
Go be a subgroup of finite index h in G. If si9 . . . , sh are a system of representatives of
the right cosets of GO in G, show that F0 — (J St • F is a Go-tessellation.
l€f<lr
(e) Same hypotheses as in (a). Let/^ 0 be a /u,-integrable function on X. Show that
there exist two points a, b in X such that
f /(*) dp,(x) and ^(P) £ f(s - b) £ f /(
JX see Jx
(*)
(Observe that if g ^ 0 is an integrable function and E an integrable subset of X, then
there exists c e E such that g(x) dp,(x) ^ #(c)/x(E), and c' G E such that
2. PARTICULAR CASES AND EXAMPLES
(14.2.1) On the additive group R, Lebesgue measure (13.1.4) is a Haar
measure (left and right, since R is commutative). This follows from the
formula for change of variable (8.7.4) applied to the function (p(£) = £ + a,
which gives f + °°/(f + a) dt = f+ °°/(0 dt for all/e Jfc(R) and all a e R.
J — oo J -~ oo
(14.2.2) Now consider the multiplicative group R* of real numbers >0.
This is a locally compact commutative group ((13.18.4), (4.1.2), and (4.1.4))
For each function fe JTC(R*), there exists a compact interval [a, b] with
0 < a < b, containing the support of /. Hence for each interval [c, d] in
R* containing Supp(/), the integral J (/(O dt)/t is defined and its value,
r 4* oo
which we denote by (/(O dt)/t, is independent of the choice of interval
/* + OO
[c, d] containing Supp(/). We assert that /K»| (f(t) dt)ft is a Haar
measure on R *. From the formula for change of variable (8.7.4), it follows
that, for each s > 0, D in R such that